A technique for analyzing past records of price change in a financial product or its derivatives and for stochastically obtaining a price distribution or a risk distribution is generally called a financial engineering technology.
In general, the Wiener process is used to model a change of stock price in the conventional financial engineering technology. The Wiener process is a type of the Markov process, which is a stochastic process on condition that a future state is independent of a past process. The Wiener process is often used to describe the Brownian motions of gas-molecules.
With variables of t (time) and z depending on the Wiener process, the Wiener process is characterized in the following relationship between Δt and Δz that is an infinitesimal change in z during the infinitesimal time Δt.Δz=ε√{square root over (Δt)}  (1)here ε is the random sample from the standard Gaussian distribution.
Thus, the Wiener process evaluates fluctuations with the variables based on the standard Gaussian distribution.
The conventional risk evaluation method for a financial product or its derivatives generally establishes upon applying the Ito's process, which is developed from the Wiener process. The Ito's process adds a drift term to the Wiener process on the assumption that a change of stock price follows the Wiener process, and further introduces a parameter function of time and other variables.
The price change in stock priced expressed by the Ito's process isdS=rSdt+Sσ√{square root over (dtW)}  (2)here S is the stock price, r is the non-risky interest rate, σ is the volatility (i.e., the predicted change rate), and W is the normal distribution with the expectation value of zero (0) and the standard deviation of one (1).
The simplest example of the Ito's process is the geometric Brownian motion model of stock prices. With the geometric Brownian motion, equation (2) becomes
                              ⅆ          x                =                                            (                              r                -                                                      σ                    2                                    2                                            )                        ⁢                          ⅆ              t                                +                      σ            ⁢                                          ⅆ                t                                      ⁢            W                                              (        3        )            here x is the natural logarithm of the stock price S.
The probability density function P(x; t) of x based on equation (3) is
                                          ∂            P                                ∂            t                          =                                            -                              (                                  r                  -                                                            σ                      2                                        2                                                  )                                      ⁢                                          ∂                P                                            ∂                x                                              +                                    1              2                        ⁢                          σ              2                        ⁢                                                            ∂                  2                                ⁢                P                                            ∂                                  x                  2                                                                                        (        4        )            
Equation 4 is the Fokker-Plank equation, and is a typical diffusion problem. The solution of equation (4) is
                              P          ⁡                      (                          x              ;              t                        )                          =                              1                                          2                ⁢                                  πσ                  2                                ⁢                t                                              ⁢                      exp            [                          -                                                (                                      x                    -                                                                  (                                                  r                          -                                                                                    σ                              2                                                        2                                                                          )                                            ⁢                                              t                        2                                                                              )                                                  2                  ⁢                                      σ                    2                                    ⁢                  t                                                      ]                                              (        5        )            and the probability density function P(x; t) of x becomes the Gaussian distribution.
Equation (5) is characterized by not only its simple form, but also effectiveness in evaluating price changes for financial products, because it is known that the price change of derivatives derived from the underlying assets has the same shape as that of the underlying assets (Ito's theorem). For this reason, various financial derivatives have been reproduced.
However, the conventional technique for evaluating risks for the financial products or the derivatives is not capable of providing sufficiently reliable results, as is known in this field.
This is because that conventional method evaluates risks of financial products based on the Gaussian distribution, and therefore, the probability of occurrence of a big price change is underestimated.
Although the likelihood of occurrence of a big price change is low, such a big price change has a significant influence to investing risks, as compared with h situations under the normal price changes. Accordingly, any risk evaluation methods or systems for financial products can not be reliable in the practical aspect unless the probability for the big price changes is accurately treated.
Another problem is that the conventional risk evaluation technique requires some corrections to a heterogeneous problem, in which the probability density function changes depending on prices, or to a non-linear problem, in which the probability density function used for the evaluation is a non-linear function. Along with the conventional approaches, such corrections have to be added empirically or based on know-how. In other words, the conventional technique requires dealer's experiences or uncertain judgements in the actual market trading.
Furthermore, the conventional risk evaluation technique has very limited capabilities for description, definition, and selection of the variables to produce price fluctuations observed in the markets. In other words, with the conventional technique, the probability density function can not be sufficiently evaluated with variables for describing risks for financial products, if the actual price change distribution of an financial product is located out of the standard Gaussian-type distribution. This insufficiency can also be true in the cases where the price change rate is influenced by the past price change rate, and correlations exist between the probability for price-up and the probability for price-down, or between the price change rate and the price change direction. The conventional technique is not capable of describing the probability density function for the price change direction as well, and therefore, the probability distribution of the price change direction for the financial products are disregarded.
Further problems in actual application of the conventional risk evaluation method for financial products relate to insufficient numerical techniques, such as random number sampling and variance reduction areas for making Monte Carlo calculations. Consequently, undesirable variance inevitably remains in the conventional technique.
Meanwhile, dealers or traders use dealing systems in banks or security companies for purposes of supporting the transactions. The conventional dealing systems calculate theoretical prices of financial products or its derivatives (hereinafter, referred to as “options”), and simulate risk evaluation or position change based on accepted theories, such as the Black-Sholes model or its expanded models. The Black-Sholes model assumes that the probability distribution of, for example, a stock price at a feature point of time is the Gaussian (or normal) distribution (Black, F. & M. Sholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 81 (May-June 1973), pp. 637-59).
However, the conventional dealing systems have many problems listed below in item (1) through (8).
(1) A so-called Fat-Tail problem is a serious problem in the financial field (Alan Greenspan, “Financial derivatives”, Mar. 19, 1999; http://www.federalreserve.gov/boarddocs/speeches/1999/19990319.htm).
In order to calculate theoretical prices for financial products or options or to simulate risk evaluation and position change, the usage of the Gaussian distribution in the conventional financial engineering models has facilitated evolution of theoretical financial engineering and implementation to a computer system easy-applicable in business areas. In other words, non-normality of probability distributions have been often observed financial markets, in which big price changes actually occur, or in which transactions are not so active. Once if this fact would be introduced from the early state, evolution theoretical financial engineering deployment and implementation to a computer system would have become much more difficult in actual application. For this reason, dealers have to transact relying upon their own experiences or intuitions. To carry out such transactions, it is critically important for the dealers to accurately grasp the behaviors for volatility of the market.
(2) There is volatility defined as a historical volatility that is calculated by the observed fluctuation of price under asset transactions. A general method for obtaining a historical volatility is to calculate the standard deviation of returns for the asset, based on the observed fluctuation of the closing price. Other known methods for calculating a historical volatility include so-called “Extreme Value Theory” for estimating a volatility from daily high prices and low prices, and a modified Parkinson method for estimating a volatility while taking discontinuity of time in the actual transaction into account.
However, these methods have drawbacks in actual application for some reasons. For example, if transaction is not active enough, there is no continuity in the movement for the price of the underlying assets. In case that the closing prices are used for determining the historical volatility, the corresponding exercise time would be altered from the actual one.
Even in the case that the transaction is active, the conventional methods are not suitable to estimate the volatility under the fat-tailed regime because these methods assume normality in the risk probability distribution for the market behaviors. To this end, the volatility calculated by the conventional methods is used only as a rough guideline within the limited applicability.
(3) Implied volatility (abbreviated as “IV”) is known in the option market, other than the historical volatility mentioned above. Implied volatility is volatility calculated back from the option prices observed in the market along with the Black-Sholes equation. Implied volatility is often used as a factor for calculating the theoretical price of an option.
However, in a non-active market (for instance, the option transaction market for underlying assets in the current Japanese security market), as the number of observed transactions for options is small, the implied volatility for the corresponding options cannot be well-defined from the actual market data. For this reason, dealers have to repeatedly manipulate factors for regular adjusting the volatility parameters through their own experiences and expert-judgements in order to reflect the current market behaviors in their dealing activities.
(4) Implied volatility of an individual stock option can provide some kind of important information to know what is the market reaction to the volatility of a particular stock for dealers, while the market reaction varies as transactions go on reflecting market circumstances.
(5) In the option market, different sets of the implied volatility are obtained for multiple options originated from the same original underlying asset. This phenomenon is referred to as a “smile effect”. In this case, approximated smile curves are drawn in the two-dimensional phase space with the combinations between a vertical axis showing the magnitude of implied volatility and a horizontal axis showing the exercise prices for the options. Those smile curves are often used to calculate theoretical prices of the option.
A volatility matrix, which is a data table having a time dimension along maturity of the option, in addition to the two-dimensional phase space mentioned above, provide information to obtain the theoretical exercise price and to interpolate the volatility value for the regions unobserved in the market behaviors up to maturity, in conjunction with the above-mentioned item (4).
(6) In fact, in order to obtain reliable smile curves or a volatility matrix, sufficient numbers of option prices must be observed in the market. On the other hand, in some cases that under the transactions in moderately active market, the option prices observed in the market are scattered in a wide range, it becomes difficult to grasp a comprehensive trend.
(7) In general, the amount of information observed in the market trades off the rationality needed in the basic assumption of the required model for estimating theoretical price for the options relating to the underlying asset price. If numbers of the past records for the option prices observed in the market are insufficient, a stronger assumption is required in a model used to obtain dynamics of an expected probability distribution for the underlying asset price. As one of the advanced models having strong assumptions, the stochastic volatility model (SVM) (Hull, John C. & Allan White, “The Pricing of Options on Assets with Stochastic Volatilities”, Journal of Finance, 42, June, 1987, pp. 281-300), and the GARCH model (T. Bollerslev, “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, Vol. 31, 1986, pp. 307-327) are well-known. However, because these models assume normality in the probability distributions, they can not deal with the Fat-Tail problem sufficiently.
There is a technique expanded from a lattice method and having no assumption of normality (Rubinstein, Mark, “Implied Binomial Trees”, Journal of Finance, 49, July 1994, pp. 771-818). This technique is capable of forming a flexible probability distribution taking the smile effect into account. However, this technique requires sufficient numbers of option prices observed in the market in order to determine the distribution form. Therefore, this technique is not suitably used in a non-active option market.
There is also a famous model named a jump model, which independently generates a stochastic process entirely different from the normal distribution for a Fat-Tail problem (e.g., R. C. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous”, Journal of Financial Economics, Vol. 3, March 1976, pp. 125-144). However, the jump model has an assumption of discontinuous price fluctuation, and therefore, the stochastic volatility model (SVM) naturally becomes a nonlinear problem. For this reason, the risk-neutral measure can not be achieved invariably, which prevent the option price from being defined uniquely.
(8) In conclusion, no conventional techniques can provide minute and accurate information in real time for solving the Fat-Tail problem and being applicable to a non-active market, although it has been desired for dealers and traders to receive significant smile curves or a volatility matrix on their displays in real time in response to the actual market that changes every moment. The conventional technique is incapable of automatically acquiring necessary data required for computation in response to requests from the dealers in an interactive manner, and of automatically selecting the optimum model to analyze the market deeply and flexibly.
Therefore, an object of the present invention is to introduce a probability density function with a higher accuracy in comparison with the normal distribution, and to develop a system capable of correctly evaluating the price distribution and the risk distribution for a financial product or its derivatives.
Another object of the present invention is to provide a price and risk evaluation system for a financial product or its derivatives, which system is capable of theoretically solving the above-mentioned heterogeneous or nonlinear problems.
It is still another object of the present invention to provide a price and risk evaluation system for a financial product or its derivatives, which system introduces a new function of probability density for estimating a price distribution and a risk distribution. This probability density function model can adequately define and describe variables that can not be dealt sufficiently by the conventional technique, and can establish a reliable method.
It is still another object of the present invention to provide a price and risk evaluation system for a financial product or its derivatives, which system introduces a new function of probability density for estimating a price distribution and a risk distribution of a financial product. This function is capable of establishing a sampling method for improving the efficiency of computation, and allows risk prices to be computed at a high efficiency.
It is yet another object of the present invention to provide a price and risk evaluation system for a financial product or its derivatives, which system is applicable to a parallel computing with high efficiency.
It is yet another object of the present invention to provide a computer-readable recording medium storing a dealing program, which includes a Boltzmann model computation developed by the nuclear reactor theories and applied to the financial field, in place of the general theories used in the conventional techniques. This program is capable of dealing with big price changes in the underlying assets (a fat-tail problem mentioned above), and is applicable to an option market in which transactions are not so active. This program allows a computer system to display significant theoretical prices and risk parameters on display terminals of dealers and traders by means of the interactive screen interfaces.